Euclid’s lemma proof

We have $a|bc$, so $bc=na$, with $n$ an integer. Dividing both sides by $a$, we have

$$\frac{bc}{a}=n$$

But $\gcd (a,b)=1$ implies $b/a$ is only an integer if $a=1$. So

$$\frac{bc}{a}=b\frac{c}{a}=n$$

which means $a$ must divide $c$.

Note that this proof relies on the Fundamental Theorem of Arithmetic. The alternative proof of Euclid’s lemma avoids this.